We study the evolution of pluri-anticanonical line bundles along the Kähler Ricci flow on a Fano manifold . Under some special conditions, we show that the convergence of this flow is determined by the properties of the pluri-anticanonical divisors of . For example, the Kähler Ricci flow on converges when is a Fano surface satisfying or . Combined with the works in [CW1] and [CW2], this gives a Ricci flow proof of the Calabi conjecture on Fano surfaces with reductive automorphism groups. The original proof of this conjecture is due to Gang Tian in [Tian90].
Cite this article
Xiuxiong Chen, Bing Wang, The Kähler Ricci flow on Fano manifolds (I). J. Eur. Math. Soc. 14 (2012), no. 6, pp. 2001–2038DOI 10.4171/JEMS/353